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14 TAYLOR'S FORMULA 193
interval I c= R into E, and J an open interval containing I, there exists an indefinitely
differentiable mapping / of R into E which coincides with g in I and with 0 in R — J. 5. Let / be a mapping of an interval I c: R into a Banach space E, and suppose / is n times differentiable at a point a e I. Show that |
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lim f/tf) -/(«) -/'(«)
,< *«, ««i\
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(use induction on n and (8.5.1) with <p(£) = (£ — a)"'1).
6. Let I c R be an interval containing 0, / an n — 1 times differentiable mapping of I
into a Banach space E. Write |
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which defines/, in I — {0}.
(a) Show that if /is n + p times differentiable at / = 0, fn can be continuously extended
to I and becomes a function which is n -}- p — \ times differentiable at all points t ^ 0 in a neighborhood V of 0 in I, and p times differentiable at t = 0; furthermore /J(0) = |
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for 0^/c^^, and lim /<P+*W = 0 for
t-.o,t*o,t6V
(Express the derivatives of/, with the help of the Taylor developments (Problem 5) of
the derivatives of/, and use Problem 2 of Section 8.6.)
(b) Conversely, let g be an n -j- p — I times differentiable mapping of I — {0} into E,
such that lim g(p+k\t)tk exists for 0 ^ k ^ n — 1. Show that the function g can
t-*0, titO, t e I
be extended to a p — 1 times differentiable mapping of I into E, and that the function
g(t)tn is n + p — 1 times differentiable in I; if furthermore #<p)(0) exists, then g(t)tn is n •+- p times differentiable at 0.
(c) Suppose I = ] — 1, 1 [, and suppose/is even in I, i.e./(— f) = /(/). Show, using (a)
and (b), that if / is In times differentiable in I, there exists an n times differentiable mapping h of I into E such that/(0 = h(t2).
7. (a) Let /be an indefinitely differentiable mapping of R" into a Banach space E. Show
that
/(0, . . . , 0) + *i/i(*i, . . . , xn) + x2 /2(*2 ,.-..,*„) H ---- + x» fn(xn)
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where /fc is indefinitely differentiable in R"-^1 (1 ^ k ^ n). Write /(*! ,..., xn) =
(/C*i, ...,*»)- /(O, x2 , . . . , *„)) 4-/(0, jca , - . - , Jf«) and apply (8.14.2) to the first summand, considered as a function of xi ; with a suitable value of p (depending on k), this will prove that (f(xlf ..., xn)— /(O, x2t ..., xn))/xi is /: times differentiable at (0, . . . , 0); finally, use induction on n.)
(b) Deduce from (a) that for any p > 0,
/<*)=> E *!1... *?/.(*)
lal^P
where all the/a are indefinitely differentiable and/0W =/(0, . . . , 0).
(c) Let / be an indefinitely differentiable mapping from R" into R; suppose that
/(O) = 0, Df /(O) = 0 for I =^ i ^ «, and that the quadratic form |
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